Statics, Dynamics, and Rheological Properties of

Micellar Solutions by Computer Simulation

Thèse en cotutelle avec l’

U

niversité

L

ibre de

B

ruxelles

THÈSE

présentée à

l’Université Paul Verlaine-Metz

par

Chien-Cheng HUANG

pour l’obtention du grade de Docteur

Spécialité : Physique

Soutenue le 25 septembre 2007

devant la commission d’examen

M. Wim

Briels

Professeur

,

Twente University, Pays-Bas

Rapporteur

M. Roland G. Winkler

Professeur, Forschungszentrum Jülich, Allemagne Rapporteur

M. Michel Mareschal

Professeur

,

Université Libre de Bruxelles

Examinateur

M. Joachim Wittmer

D

R CNRS

,

Institut Charles Sadron, Strasbourg

Examinateur

Mme

Hong Xu

Professeur

,

Université Paul Verlaine-Metz

Directeur de Thèse

M. Jean-Paul Ryckaert

Professeur

,

Université Libre de Bruxelles

Directeur de Thèse

Remerciements

Dans le cadre d’une thèse en co-tutelle entre le Laboratoire de Physique des Milieux

Denses de l’Université Paul Verlaine-Metz et le Laboratoire de Physique des

Polymères de l’Université de Libre de Bruxelles, j’ai été amené à travailler au sein de

deux laboratoires, ce qui fut une expérience très enrichissante tant d’un point de vue

scientifique qu’humain.

J’exprime, tout d’abord, ma reconnaissance envers le Prof. Hong Xu pour sa

pédagogie, sa grande disponibilité et son soutien ainsi que pour la formation qu’elle

m’a dispensé. Je tiens à remercier ensuite le Prof. Jean-Paul Ryckaert pour toutes les

discussions productives que nous avons eues ensemble et le partage de son savoir

dans le domaine de la physique. Je les remercie pour leur soutien jour après jour et

pour leur qualité humaine.

Merci à vous deux de m’avoir donné un si bel exemple de rigueur et d’honnêteté

scientifique.

Je remercie également le Dr. Joachim Wittme et le Prof. Marc Baus pour les

nombreuses idées sur les voies à explorer ainsi que pour les discussions très

enrichissantes.

Je remercie grandement Monsieur Georges Destrée pour les aides sur les techniques

de simulations.

Je souhaite aussi remercier Monsieur François Crevel pour les nombreuses

discussions.

Je voudrais également remercier l’ensemble des membres du jury de thèse, et en

particulier le Prof. Roland G. Winkler et le Prof. Wim Briels pour avoir accepté la

charge de rapporteurs de cette thèse. Merci au Prof. Michel Mareschal d’avoir bien

voulu présider le jury et Dr. Joachim Wittme pour l’intérêt qu’il a porté à ce travail.

Enfin, je tiens également à remercier ma famille et mes proches pour leur confiance et

leur soutien constant.

Abstract

Statics,dynamics,rheology and scission recombination kinetics of self as-

sembling linear micelles are investigated at equlibriumstate and under shear

ﬂowby computer simulations using a newly proposed mesoscopic model.We

model the micelles as linear sequences of Brownian beads whose space-time

evolution is governed by Langevin dynamics.A Monte Carlo algorithm

controls the opening of a bond or the chain-end fusion.A kinetic param-

eter ω modelling the eﬀect of a potential barrier along a kinetic path,is

introduced in our model.For equilibrium state we focus on the analysis

of short and long time behaviors of the scission and recombination mech-

anisms.Our results show that at time scales larger than the life time of

the average chain length,the kinetics is in agreement with the mean-ﬁeld

kinetics model of Cates.By studying macroscopic relaxation phenomena

such as the average micelle length evolution after a T-jump,the monomer

diﬀusion,and the zero shear relaxation function,we conﬁrm that the ef-

fective kinetic constants found are indeed the relevant parameters when

macroscopic relaxation is coupled to the kinetics of micelles.For the non-

equilibrium situation,we study the coupled eﬀects of the shear ﬂow and the

scission-recombination kinetics,on the structural and rheological properties

of this micellar system.Our study is performed in semi-dilute and dynam-

ically unentangled regime conditions.The explored parameter ω range is

chosen in order for the life time of the average size chain to remain shorter

than its intrinsic (Rouse) longest relaxation time.Central to our analysis

is the concept of dynamical unit of size Λ,the chain fragment for which the

life time τ

Λ

and the Rouse time are equal.Shear thinning,chain gyration

tensor anisotropy,chain orientation and bond stretching are found to de-

pend upon the reduced shear rate β

Λ

= ˙γτ

Λ

while the average micelle size

is found to decrease with increasing shear rate,independently of the height

of the barrier of the scission-recombination process.

Contents

Introduction 1

1 Theoretical Framework 7

1.1 Statistical mechanics derivation of the distribution of chain lengths...7

1.2 A kinetic model for scissions and recombinations.............13

1.2.1 Macroscopic thermodynamic relaxation..............16

1.3 Linear viscoelasticity of unentangled micelles...............17

1.3.1 The Rouse model...........................18

1.3.2 The Rouse model implications on the viscous response of a monodis-

perse polymer solution........................19

1.3.3 The theory of Faivre and Gardissat and the viscoelasticity of

micelles................................20

2 The mesoscopic model of worm-like micelles 23

2.1 The potential [19]...............................24

2.2 The Langevin Dynamics...........................25

2.2.1 Nonequilibrium LD technique....................28

2.3 Monte-Carlo procedure............................29

3 Equilibrium Properties 35

3.1 Static properties...............................36

3.1.1 List of simulation experiments and chain length distribution...36

3.1.2 Chain length conformational analysis................38

3.1.3 Pair correlation function of chain ends and the distribution of

bond length..............................40

3.2 Dynamic properties..............................42

i

3.2.1 Kinetics Analysis...........................43

3.2.2 Analysis of the Monomer Diﬀusion.................65

3.2.3 Zero-shear stress time autocorrelation functions..........71

3.2.4 Macroscopic relaxation behavior..................73

3.3 General comments..............................74

4 Non-equilibrium properties 77

4.1 Collective rheological behavior.......................78

4.1.1 Orientation of the chains......................78

4.1.2 Viscosity...............................82

4.2 Chain length distribution and chain size dependent properties......85

4.2.1 Eﬀect of shear ﬂow on distribution of chain lengths........85

4.2.2 Saturation eﬀect on orientational properties............87

4.3 Scission-recombination kinetics under ﬂow.................92

5 Conclusions 99

Bibliography 105

ii

Introduction

In the ﬁeld of self-assembling structures,supramolecular polymers are attracting

nowadays much attention [1,56].The terminology of supramolecular polymer applies

to any polymer-like ﬂexible and cylindrical superstructure obtained by the reversible

linear aggregation of one or more type of molecules in solution or in melt.These

supramolecular polymers are typical soft matter systems and the chain length dis-

tribution which determines their properties,is very sensitive to external conditions

(temperature,concentration,external ﬁelds,salt contents,etc...).Wormlike micelles

are one of the most common type of supramolecular polymer [2].Self-assembled stacks

of discotic molecules [20] and chains of bifunctional molecules [21] are other examples.

All these examples diﬀer by the nature of the intermolecular forces involved in the self-

assembling of the basic units,but they lead to a similar physical situation bearing much

analogy with a traditional system of polydisperse ﬂexible polymers when their length

becomes suﬃciently large with respect to their persistence length.The speciﬁcity and

originality of these supramolecular polymers comes from the fact that these chains are

continuously subject to scissions at random places along their contour and subject to

end to end recombinations,leading to a dynamical equilibrium between diﬀerent chain

lengths species.

It is well known and schematically illustrated in ﬁgure 1 that some surfactant

molecules in solution can self-assemble and form wormlike micelles [2].As their mass

distribution is in thermal equilibrium,they are therefore sometimes termed “equilib-

rium polymers” (EP) [1].Similar system of equilibrium polymer are formed by liquid

sulfur [33,34,35],selenium [36] and some protein ﬁlaments [37]

The micellar solutions exhibit fascinating rheological behavior which has been re-

cently reviewed and discussed theoretically [56].Quite commonly,under shear ﬂow

in a Couette cell,an originally isotropic micellar solution undergoes a shear-banding

1

transition [3],producing a two phases system spatially organized in a concentric man-

ner with both phases lying either close to the inner or to the outer cylinder of the

rheometer.As the viscosity of both phases is diﬀerent,a velocity proﬁle with two

slopes is observed through the gap.This non-equilibriumphase separation is the object

of numerous studies [56].The shear thinning and orientational ordering of wormlike

micelles [55] resembles the usual phenomenon observed in polymer solutions but,in

many micellar solutions close to the overlap concentration[4],one can also observe a

shear-thickening behavior whose microscopic origin is still a matter of debate nowa-

days.Much better understood is the trend of entangled micellar solutions to display a

Maxwell ﬂuid rheological behavior [2].The simple exponential relaxation behavior has

been theoretically explained by the reptation-reaction model[56],taking into account

the scission-recombination mechanism by which local entanglements can be released by

chain scission.The theoretical treatment of the scission-recombination process within

the rheological theoretical approaches,is usually based on a simple mean-ﬁeld (MF)

approach in which correlations between successive kinetic events are fully neglected.

The way to take into consideration such correlated transitions has been detailed by

O’Shaughnessy and Yu [12] in order to explain some high frequency features of the rhe-

ological behavior of micellar systems.As the latter kinetics model interprets the stress

relaxation as the result of local eﬀects between successive correlated transitions where

chain ends produced by a scission recombine with each other after a small diﬀusive ex-

cursion of the chain ends,such a rheological behavior was called “diﬀusion-controlled”

(DC) by opposition to the standard MF model.

Computer simulations at the mesoscopic scale oﬀers an interesting route towards

the understanding of the generic properties of wormlike micelles solutions on the basis

of a well controlled microscopic (or mesoscopic) model.The ﬁrst accent within the

simulation approach was put on testing the chain length distribution for various algo-

rithms producing,on a lattice or in continuum space,temporary linear self assembling

structures.Earlier Monte Carlo simulations using an asymmetric Potts model were

performed to study static properties of equilibrium polymers [38,39,40].Rouault and

Milchev proposed a dynamical Monte Carlo algorithm [41],based on the highly eﬃcient

bond ﬂuctuation model (BFM) [42,43].In great detail,Wittmer and co-workers [13,22]

investigate the static properties of EP in dilute and semi-dilute solutions using the

same BFM.They conﬁrmed scaling predictions of the mean chain length in dilute and

2

semi-dilute limits.Rouault [15] veriﬁed the dependence of mean chain length with con-

centration using an oﬀ-lattice Brownian dynamics simulation.Dynamics at equilibrium

and direct simulation of systems in shear ﬂow were also much investigated over the last

ten years.Kroger and co-workers study EP at equilibrium [58] and under shear ﬂow

[17] by Molecular Dynamics using the particular pair potential FENE-C model.This

model is a variant of the traditional pair potential of the LJ+FENE type used to model

(permanent) polymers,a pair potential which diverges both as r goes to zero (repulsive

forces) and at a distance r = R

c

where the corresponding attractive force between

neighboring monomers also diverges.The FENE-C potential is a truncated form of the

usual form which is set to a constant beyond a certain distance r = r

m

< R

c

value,

creating a ﬁnite potential well for a bounded pair with respect to the unbounded pair

(r > r

m

).In this way,the covalent bond can break or recombine as the pair distance

crosses the r

m

value.Using the same model,Padding and Boek [16] investigated the

recombination kinetics and the stress relaxation of wormlike micellar systems.They

found that at high concentrations,the kinetics is close to a diﬀusion-controlled mecha-

nism.Milchev and co-works [26] study micelle conformations and their size distribution

by an oﬀ-lattice microscopic model,to study solutions of EP in a lamellar shear ﬂow

while Padding and Boek investigate the inﬂuence of shear ﬂow on the formation of

rings in a EP system using FENE-C model [44].All these studies predict a decrease of

the average micelle size as a result of the shear ﬂow.Other simulation studies envisage

that ultimately,various simulations at diﬀerent length and time scale will have to be

combined to ﬁll the huge gap between the atomistic length and time scales where the

precise chemistry is relevant and the mesoscopic scale where rheological properties can

really be probed by simulation.Studies of this type mixing both the atomistic and

mesoscopic approaches to study wormlike micelles rheology are under progress[57].

The aim of our thesis on isotropic wormlike micelles solutions is to analyze,on the

basis of a dynamical simulation at the mesoscopic level,the inﬂuence on the macro-

scopic relaxation phenomena of the dynamical coupling between the usual “ﬂexible

polymer” relaxation processes and the “scission-recombination” kinetics.We will re-

strict ourselves to unentangled solutions (working slightly below or above the isotropic

semi-dilute threshold with a very ﬂexible mesoscopic equilibrium polymer model) to

avoid the prohibitive computer time needed to follow relaxation phenomena governed

by entangled dynamics.The relevance of the MF kinetics model and the microscopic

3

origin of the scission-recombination rate constants is one important target of our study.

Unentangled supramolecular polymers dynamics was indeed found to be relevant in

selenium rheology.At the occasion of this experimental rheological study,Faivre and

Gardissat [36] proposed an extension of the traditional Rouse theory to include the

scission kinetic eﬀects on the Rouse modes relaxation.They predicted the way the

zero shear rate viscosity decreases as the scission and recombination kinetic constants

increase,introducing the concept of dynamical subunit whose size ﬁxes the relaxation

time which governs the shear modulus relaxation.

The thesis is organized as follows:The theoretical aspects of wormlike micelles

statics (distribution of micelle sizes) and the dynamics (Cates MF kinetic model,Faivre

and Gardissat theory[36])are gathered in chapter I.

In chapter II,we detail the particular mesoscopic Langevin Dynamics model which

is adopted throughout our thesis.Given our aims,this model has the particularly

useful feature that the static properties (in particular the distribution of chain lengths)

and the dynamics of the scission-recombination can be tuned separately so that we

will often investigate how the relaxation at a unique state point (from a static point of

view) is modiﬁed by tuning the kinetic rate constants.

Chapter III reports the results of a series of simulations at equilibrium,performed

by Langevin Dynamics.Both a dilute and a semi-dilute state points are treated.We

check our results regarding the distribution of chain sizes with respect to theoretical

prediction and analyze in detail the microscopic origin of the relevant rate constants.

Chapter IV reports a systematic study of a single semi-dilute state point of micellar

solution under shear ﬂow.Here two kinds of parameters,the shear rate and the scission-

recombination rate constants,are systematically varied.We discuss the evolution of

the average micelle size with shear rate and relate it to a diﬀerent evolution of the two

types of rate constants (scission and recombination).We also discuss the nature of

the relevant reduced shear rate and extrapolate our viscosity data to zero shear rate,

allowing us to test the Faivre-Gardissat theory’s predictions [36].

Our general conclusions are gathered in the last chapter V.

4

Cylindrical micelles

Equilibrium polymer

COARSE−GRAINING

Surfactant

k

k

s

r

B

q

Chain end

ScissionRecombination

E

Figure 1:Some surfactant molecules in solution self-assemble and form long wormlike

micelles which continuously break and recombine.Their mass distribution is,hence,in

thermal equilibriumand they present an important example of the vast class of systems

termed “equilibrium polymers”[1].The free energy E of the (hemispherical) end cap of

these micelles has been estimated [2] to be of order of 10k

B

T.This energy penalty

(together with the monomer density) determines essentially the static properties and

ﬁxes the ratio of the scission and recombination rates,k

s

and k

r

.Additionally,these

rates are inﬂuenced by the barrier height B which has been estimated to be similar to

the end cap energy.Both important energy scales have been sketched schematically as

a function of a generic reaction coordinate q (see chapter 8 of reference [23]).Following

closely the analytical description [2,22,13] these micellar systems are represented in

this study by coarse-grained eﬀective potentials in terms of a standard bead-spring

model.The end cap free energy becomes now an energy penalty for scission events,

i.e.,the creation of two unsaturated chain ends.The dynamical barrier is taken into

account by means of an attempt frequency ω = exp(−B/k

B

T).If ω is large,suc-

cessive breakage and recombination events for a given chain can be assumed to be

uncorrelated and the recombination of a newly created chain ends will be of standard

mean-ﬁeld type.On the other hand,the (return) probability that two newly created

chain ends recombine immediately must be particularly important at large ω.These

highly correlated “diﬀusion controlled” [12] recombination events do not contribute to

the eﬀective macroscopic reaction rates which determine the dynamics of the system.

6

Chapter 1

Theoretical Framework

In this chapter we will brieﬂy present the theoretical framework of cylindrical micelles

and some standard polymer theoretical aspects which are pertinent to the ﬂexible

micelles system.In Section 1,we introduce the statistical mechanics derivation of the

distribution of chain lengths and of the corresponding average chain length.Then in

section 2,for the scission-recombination kinetics at equilibrium,we study the theoretical

formalismof equilibriumpolymers (EP) developed by Cates and co-workers [9,2],based

on a mean-ﬁeld approximation.And the third section of this chapter will focus on the

linear viscoelasticity of EP.In this section,a framework for linear viscoelacity of dilute

polymer solutions,and intrinsic shear modulus by the Rouse model are presented.

Finally,we brieﬂy review the theory of Faivre and Gardissat [36] where a modiﬁcation

of the standard Rouse theory of linear viscosity of a polydisperse polymer system is

proposed.

1.1 Statistical mechanics derivation of the distribution of

chain lengths

To treat a system of wormlike micelles theoretically,it is convenient to work at the

mesoscopic scale using a model of linear ﬂexible polymers made of L monomers of size

b linked together by a non permanent bonding scheme.Within the system,individual

chain lengths ﬂuctuate by bond scission and by fusion of chain ends of two diﬀerent

chains.Statistical mechanics can be employed to predict the equilibrium distribution

of chain lengths[2].In terms of the equilibriumchain number density c

0

(L),the average

chain length L

0

and the total monomer density φ are given by

7

L

0

=

P

∞

L=0

Lc

0

(L)

P

∞

L=0

c

0

(L)

(1.1)

φ =

∞

X

L=0

Lc

0

(L) =

M

V

(1.2)

where M denotes the total number of monomers in the system and V is the volume.

Conceptually,we consider the Helmholtz free energy F(V,T;{N(L)},N

s

) of a mix-

ture of chain molecules of diﬀerent length L in solvent where,in addition to the temper-

ature T,the volume V and the number of solvent molecules N

s

,the number of chains

of each speciﬁc length N(L) is ﬁxed.Let F(V,T;M,N

s

) be the Helmholtz free energy

of a similar system where only the total number of solute monomers M is ﬁxed.The

equilibrium chain length distribution c

0

(L) = N(L)/V will result from the set {N(L)}

which satisﬁes the condition

F(V,T;M,N

s

) = min

{N(L)}

"

F(V,T;{N(L)},N

s

) +

∞

X

L=0

LN(L)

#

(1.3)

The parameter is the Lagrange multiplier associated with the constraint that

the individual number of chains N(L) must keep ﬁxed the total number of monomers

M =

P

∞

0

LN(L).Minimization requires that the ﬁrst derivative with respect to any

N(L

′

) variable (L’=1,2,...) is zero,giving

δF(V,T;{N(L)},N

s

)

δN(L

′

)

+L

′

= 0;L

′

= 1,2,...(1.4)

We expect the entropic part of the total free energy F(V,T;{N(L)},N

s

) to be the

sumof translational and chain internal conﬁgurational contributions which both depend

upon the way the M monomers are arranged into a particular chain size distribution.

For the translation part,the polydisperse system entropy is estimated as the ideal

mixture entropy S

id

S

id

(V,T;{N(L)},N

s

) = −k

B

X

L

N(L) ln(CN(L)) +S

solv

(1.5)

where C = b

3

/V is a dimensionless constant independent of L and where S

solv

is

the solvent contribution,independent of the N(L) distribution.The conﬁgurational

entropy of an individual chain with L monomers is written as S

1

(L) = k

B

lnΩ

L

,in

8

1.1 Statistical mechanics derivation of the distribution of chain lengths

terms of Ω

L

,the total number of conﬁgurations of the chain.Adding the conﬁgurational

contributions to S

id

as given by eq.(1.5),the total entropy becomes

S(V,T;{N(L)},N

s

) = −k

B

X

L

N(L) [ln(CN(L)) −lnΩ

L

] +S

solv

(1.6)

We now turn to the energy E(V,T;{N(L)},N

s

) of the same system.If E

1

(V,T;

L) represents the internal energy of a chain of L monomers and E

s

the energy of a

solvent molecule,the energy can be written as

E(V,T;{N(L)},N

s

) =

X

L

N(L)E

1

(V,T;L) +N

s

E

s

(V,T) (1.7)

The key contribution in E

1

is the chain end-cap energy E which corresponds to

the chain end energy penalty required to break a chain in two pieces.We will suppose

that E

1

(L) = E + L˜ǫ where ˜ǫ is an irrelevant energy per monomer as M˜ǫ,its total

contribution to the system energy,is independent of the chain length distribution.

The present approximation of the total free energy of the system is thus given by

incorporating in the general expression (1.3) the expressions (1.6) and (1.7),giving

βF(V,T;{N(L)},N

s

) =

X

L

N(L) [lnN(L) +lnC −lnΩ

L

+βE +β˜ǫL] (1.8)

where irrelevant constant solvent terms have been omitted as we only need the ﬁrst

derivative of the free energy with respect to N(L),which now takes the form

δβF

δN(L)

= lnN(L) +lnC −lnΩ

L

+βE +β˜ǫL+1.(1.9)

With this expression,the minimization condition on N(L) becomes

lnN(L) +lnC −lnΩ

L

+βE +1 +

′

L = 0 (1.10)

where

′

= (β +β˜ǫ).We note at this stage that the second derivative of βF(V,T;

{N(L)}) +

′

P

L

LN(L) with respect to N(L) and N(L

′

) variables gives the non neg-

ative result

δ

LL

′

N(L)

,indicating that the extremum is indeed a minimum.

Solving for N(L) in eq.(1.10),we get

N(L) = C

′

−1

exp−(

′

L+βE −lnΩ

L

) (1.11)

where C

′

= eC while,according to eq.(1.2)),

′

must be such that

X

L

Lexp−(

′

L+βE −lnΩ

L

) = MC

′

(1.12)

9

The equilibrium N(L) variables are also related to the equilibrium chain length

average L

0

(see eq.(1.1)),so that

X

L

exp−(

′

L+βE −lnΩ

L

) =

MC

′

L

0

(1.13)

To progress,we nowneed to specify the explicit L dependence of Ω

L

.The traditional

single chain theories of polymer physics provide universal expression of Ω

L

in terms of

the polymer size,the environment being simply taken into account through the solvent

quality and the swollen blob size in the semi-dilute (good solvent) case.

1.1.0.1 The case of mean ﬁeld or ideal chains

The basic mean-ﬁeld or ideal chain model for a L segments chain gives

Ω

id

L

=

C

1

z

L

(1.14)

where z is the single monomer partition function and C

1

a dimensionless constant.

Adapting eq.(1.11),one has

N(L) =

C

1

C

′

exp−(βE) exp(−”L) (1.15)

where ” =

′

−lnz must,according to eq.(1.12),be such that

X

L

Lexp−(”L) =

1

”

2

=

MC

′

C

1

exp−(βE)

(1.16)

while eq.(1.13) takes the form

X

L

exp−(”L) =

1

”

=

MC

′

L

0

C

1

exp−(βE)

(1.17)

In eqs.(1.16) and (1.17),sums over L from 1 to ∞have been approximated by the

result of their continuous integral counterparts.

Combining eqs.(1.15),(1.16) and (1.17),one gets the ﬁnal expression for the chain

number densities

c

0

(L) =

φ

L

2

0

exp(−

L

L

0

) (1.18)

with the average polymer length given by

L

0

= B

1/2

1

φ

1

2

exp

βE

2

.(1.19)

where B

1

= eb

3

/C

1

is a constant depending upon the monomer size b and the prefactor

in the number of ideal chain conﬁgurations in eq.(1.14).

10

1.1 Statistical mechanics derivation of the distribution of chain lengths

1.1.0.2 The case of dilute chains in good solvent

Polymer solutions are in a dilute regime when chains do not overlap and in semi-dilute

regime when chains do strongly overlap while the total monomer volume fraction is still

well below its melt value.In the semi-dilute regime in good solvent condition,chains

remain swollen locally over some correlation length,known as the swollen blob size χ,

but they are ideal over larger distances as a result of the screening of excluded volume

interactions between blobs.

Speciﬁcally,for a given monomer number density φ,the blob size is given by the

condition that the blob volume χ

3

= b

3

L

∗3ν

is equal to the total volume V divided by

the total number of blobs M/L

∗

in the swollen blob.This gives estimates in terms of

the reduced number density φ

′

= b

3

φ,

L

∗

= φ

′

(

1

1−3ν

)

(1.20)

χ = bφ

′

(

ν

1−3ν

)

(1.21)

where ν = 0.588 in present good solvent conditions[8].

In living polymers characterized by a monomer number density φ and some averaged

chain length L

0

,the semi-dilute conditions correspond to the case L

0

>> L

∗

.We

discuss in this subsection the theory for the dilute case where L

0

<< L

∗

.We will come

back to the semi-dilute case in the next subsection.

Self-avoiding walks statistics apply to dilute chains in good solvent,and we thus

adopt the number of conﬁgurations[6,7](See especially page 128 of the book of Grosberg

and Khokhlov [7])

Ω

EV

L

= C

1

L

(γ−1)

z

L

(1.22)

for a chain of size L,where γ is the (entropy related) universal exponent equal to 1.165,

z is the single monomer partition function and C

1

a dimensionless constant.

Incorporating expression (1.22) in eq.(1.11),one gets

N(L) =

V

B

1

exp−(βE)L

(γ−1)

exp−(”L) (1.23)

where B was introduced in eq.(1.19) and where ” =

′

−lnz must be ﬁxed by eq.(1.2)

X

L

L

γ

exp−(”L) = B

1

φexp(βE) (1.24)

11

while eq.(1.13) takes here the form

X

L

L

(γ−1)

exp−(”L) =

B

1

φ

L

0

exp(βE) (1.25)

If L is treated as a continuous variable,eqs (1.24) and (1.25) can be rewritten in

terms of the Euler Gamma function satisfying Γ(x) = xΓ(x −1) as

Z

∞

0

L

γ

exp−(”L)dL =

Γ(γ +1)

”

(γ+1)

= B

1

φexp(βE) (1.26)

Z

∞

0

L

(γ−1)

exp−(”L)dL =

Γ(γ)

”

γ

=

B

1

φ

L

0

exp(βE) (1.27)

From eqs (1.26) and (1.27),one gets

” =

γ

L

0

(1.28)

B

1

φexp(βE) =

Γ(γ +1)

γ

(γ+1)

L

(γ+1)

0

(1.29)

These results lead then ﬁnally to the Schulz-Zimmdistribution of chain lengths,namely

c

0

(L) =

exp(−βE)

B

1

L

(γ−1)

exp(−γ

L

L

0

) (1.30)

and an average polymer length given by

L

0

=

γ

γ

Γ(γ)

1

1+γ

B

1

1+γ

1

φ

1

1+γ

exp

βE

(1 +γ)

.(1.31)

1.1.0.3 The semi-dilute case

We consider here the semi-dilute case in good solvent where the average length of living

polymers L

0

is much larger than the blob length L

∗

.The usual picture of a semi-dilute

polymer solution is an assembly of ideal chains made of blobs of size χ.Using this

approach,Cates and Candau [2] and later J.P.Wittmer et al [13] derived the relevant

equilibrium polymer size distribution.In this subsection,we adapt their derivation to

the theoretical framework presented above.

Let Ω

b

be the number of internal conﬁgurations per blob and z

′

some coordination

number for successive blobs.As there are n

b

= L/L

∗

blobs for a chain of L monomers,

we write the total number of internal conﬁgurations of a chain of size L as

Ω

SD

L

= C

1

L

∗(γ−1)

Ω

L/L

∗

b

z

′

L/L

∗

(1.32)

12

1.2 A kinetic model for scissions and recombinations

where γ is the universal exponent in the excluded volume chain statistics met earlier

for chains in dilute solutions.The important factor L

∗(γ−1)

can be seen as an entropy

correction for chain ends just like E was an energy correction to L˜ǫ.This entropic

term which involves the number of monomers per blob,is needed to take into account

that when a chain breaks,its two ends are subject to a reduced excluded volume

repulsion.The other factors in eq.(1.32) will lead to terms linear in L after taking

the logarithm and thus will be absorbed in the Lagrange multiplier deﬁnition,as seen

earlier in similar cases for ideal and dilute chains.The resulting expression of N(L) in

terms of the Lagrange multiplier (cf.eq.(1.15)) can then be written by analogy as

N(L) =

C

1

C

′

exp−[βE −(γ −1) lnL

∗

] exp(−L) (1.33)

Proceeding as in the ideal case (simply replacing at every step the constant βE

by βE − (γ − 1) lnL

∗

,one recovers in the semi-dilute case the simple exponential

distribution

c

0

(L) =

φ

L

2

0

exp(−

L

L

0

) (1.34)

with a slightly diﬀerent formula for the average polymer length

L

0

∝ φ

α

exp

βE

2

(1.35)

where α =

1

2

(1 +

γ−1

3ν−1

) is about 0.6.

1.2 A kinetic model for scissions and recombinations

The interest for wormlike micelles dynamics came from the experimental observation

that entangled ﬂexible micelles often display,after an initial strain,a simple exponential

stress relaxation.The mechanism of such relaxation is diﬀerent from that of usual

dead polymer entangled melts.In the latter system,stress relaxation requires that

individual chains leave the strained topological tube created by the entangled temporary

network by a reptation mechanism.A theoretical model,taking into account the extra

relaxation mechanismcaused by scissions and recombinations of micelles,leads [9] to an

exponential decay of the shearing forces with a decay Maxwell time equal to τ ≈

√

τ

b

τ

rep

where τ

rep

is the chain reptation time and τ

b

is the mean life time of a chain of average

size in the system.

13

We will assume in the following that the Cates’s scission-recombination model gov-

erning the population dynamics,originally devised to explain entangled equilibrium

polymer melt rheology,should also apply to the kinetically unentangled regime which

is explored in the present work.

This Cates kinetic model [9] assumes that

• the scission of a chain is a unimolecular process,which occurs with equal proba-

bility per unit time and per unit length on all chains.The rate of this reaction is

a constant k

s

for each chemical bond,giving

τ

b

=

1

k

s

L

0

(1.36)

for the lifetime of a chain of mean length L

0

before it breaks into two pieces.

• recombination is a bimolecular process,with a rate k

r

which is identical for all

chain ends,independently of the molecular weight of the two reacting species they

belong to.It is assumed that recombination takes place with a new partner with

respect to its previous dissociation as chain end spatial correlations are neglected

within the present mean ﬁeld theory approach.It results from detailed balance

that the mean life time of a chain end is also equal to τ

b

.

Let c(t,L) be the number of chains per unit volume having a size L at time t.On

the basis of the model,the following kinetic equations can be written [9]

dc(t,L)

dt

= −k

s

Lc(t,L) +2k

s

Z

∞

L

c(t,L

′

)dL

′

+

k

r

2

Z

L

0

c(t,L

′

)c(t,L −L

′

)dL

′

−k

r

c(t,L)

Z

∞

0

c(t,L

′

)dL

′

(1.37)

where the two ﬁrst terms deal with chain scission (respectively disappearance or appear-

ance of chains with length L) while the two latter terms deal with chain recombination

(respectively provoking the appearance or disappearance of chains of length L).

It is remarkable that the static solution of this empirical kinetic model leads to an

exponential distribution of chain lengths.Indeed,direct substitution of solution c

0

(L)

in the above equation leads to the detailed balance condition:

φ

k

r

2k

s

= L

2

0

(1.38)

14

1.2 A kinetic model for scissions and recombinations

the ratio of the two kinetic constant being thus restricted by the thermodynamic state.

Detailed balance means that for the equilibrium distribution c

0

(L),the number of

scissions is equal to the number of recombinations per unit time and volume.The total

number of scissions and recombinations per unit volume and per unit time,denoted

respectively as n

s

and n

r

,can be expressed as

n

s

= k

s

φ

L

2

0

Z

∞

0

Lexp(−L/L

0

)dL = k

s

φ (1.39)

n

r

=

k

r

2

φ

2

L

4

0

Z

∞

0

dL

′

Z

∞

0

dL”exp(−

L

′

L

0

) exp(−

L”

L

0

) =

φ

2

k

r

2L

2

0

(1.40)

and it can be easily veriﬁed that detailed balance condition equation 1.38 implies n

s

=

n

r

.

Mean ﬁeld theory assumes that a polymer of length L will break on average after

a time equal to τ

b

= (k

s

L)

−1

according to a Poisson process.This implies that the

distribution of ﬁrst breaking times (equal to the survival times distribution) must be

of the form

Ψ(t) ∝ exp(−

t

τ

b

) (1.41)

for a chain of average size.Detailed balance then requires that the same distribution

represents the distribution of ﬁrst recombination times for a chain end[2].Accordingly,

throughout the rest of this chapter,the symbol τ

b

will represent as well the average

time to break a polymer of average size or the average time between end chain recombi-

nations.In the same spirit,we stress that among the diﬀerent estimates of τ

b

proposed

in this work,some are based on analyzing the scission statistics while others are based

on the recombination statistics.

Two additional points may be stressed at this stage:

• The mean ﬁeld model in the present context has been questioned [12] because

in many applications,there are indications that a newly created chain end often

recombines after a short diﬀusive walk with its original partner.In that case,a

possibly large number of breaking events are just not eﬀective and the kinetics

proceeds thus diﬀerently.

• Given the statistical mechanics analysis in the previous subsection,we see that

the equilibrium distribution of chain lengths resulting from the simple empirical

kinetic model is perfectly compatible with the equilibriumdistribution in polymer

15

solutions at the θ point (ideal chains) or for semi-solutions (ideal chains of swollen

blobs).The kinetic model is still pertinent,given its simplicity,in the case of

dilute solutions in good solvent as the chain length distribution although non

exponential,is not very far from it.

1.2.1 Macroscopic thermodynamic relaxation

In equation (1.35) the average length L

0

depends on the thermodynamic variables of

the systemsuch as the temperature,the pressure,and the volume fraction of monomers

φ.Under a sudden change in one of the thermodynamic variables (for example,the

temperature),the distribution of the length of the chains in equation (1.37) relaxes

to a new equilibrium exponential distribution characterized by a new average length

L

∞

.The corresponding theory of macroscopic thermodynamic relaxation has been

given by Marques and Cates [27] and tested using a Monte Carlo study by Michev

and Rouault [46].This is in general a complicated,nonlinear decay which can be

monitored experimentally by light scattering which probes the evolution of the average

length.The theory is based on the equations of distribution of length of chain (1.34)

and the kinetic model of Cates [9] i.e.equation (1.37).The characteristic time provides

information about the kinetics of the system.

Marques and Cates [27] show that for any amplitude of the perturbation which

conserves the total volume fraction of monomers (for instance,a temperature modiﬁca-

tion having an arbitrary time-dependence),the distribution function c(t,L) of equation

(1.37) remains exponential versus L.Indeed,they show that

c(t,L) = φf

2

(t) exp{−Lf(t)} (1.42)

is a non-linear eigenfunction of equation (1.37) with a eigenvalue f(t) that obeys

df

dt

= k

s

1 −

φk

r

2k

s

f

2

(t)

,(1.43)

where k

r

and k

s

the kinetic constants of the new equilibriumstate,after,e.g.a T-jump.

The time evolution of f(t) for a thermodynamic jump is then the solution of (1.43) with

the initial condition f(t = 0) = 1/L

0

.One obtains [27]:

f(t) = L

−1

∞

tanh

t +t

0

2τ

if L

0

> L

∞

(1.44)

f(t) = L

−1

∞

coth

t +t

0

2τ

if L

0

< L

∞

(1.45)

16

1.3 Linear viscoelasticity of unentangled micelles

where the constant t

0

is given by

t

0

= 2τ tanh

L

∞

L

0

if L

0

> L

∞

(1.46)

t

0

= 2τ coth

L

∞

L

0

if L

0

< L

∞

,(1.47)

and the time

τ =

1

2k

s

L

∞

(1.48)

These equations describe the recovery of c(t,L) and hL(t)i the average chain length,

following a sudden change in temperature which is given by

hL(t)i =

1

f(t)

(1.49)

We see that by measuring the non-equilibrium τ the relaxation time of the mean chain

length,one can get an estimate of the microscopic scission rate constant k

s

.

1.3 Linear viscoelasticity of unentangled micelles

To describe the dynamics of cylindrical micelles in solution,each micelle can be mod-

elled as a linear collection of L identical fragments (L being proportional to the length

of the micelle),subject to Brownian motion to represent the dynamical eﬀect of the

bath (the coupling to the rest of the degrees of freedom),subject to connecting forces

between adjacent fragments to enforce the linear structure of the micelle and possibly

subject to pair interactions between non connected fragment pairs.If,as the rela-

tive distance between adjacent fragments increases,the connecting tension force is not

bounded to a ﬁnite value to allow an intrinsic possibility of scission (e.g.the FENE-C

model of Kroger[17]),the scission and recombination processes must be incorporated

independently in the model.The latter must specify the way through which individ-

ual bonds can break (chain scission) or reform (chain end recombinations) by a pair

potential swap for the involved pair of fragments.The micelle dynamics thus needs

to combine all these various ingredients,implying that individual chain entities will

survive for a ﬁnite life time during which ordinary polymer relaxation takes place.

Within a theoretical approach of the dynamical properties of micellar solutions

where the emphasis lies in the search for analytical predictions,it is useful to explore

the validity of the traditional Rouse model to represent the “dead polymer” dynamics

17

part (valid as long as an entity survives),when it can be assumed that excluded volume

and entanglement eﬀects are absent or weak.In the present purely theoretical discus-

sion,we thus restrict ourselves to the unentangled dynamics of θ solvent chains.We

brieﬂy review the basic ingredients and essential results of the Rouse model for “dead

chains”,including linear viscoelasticity and we report the extension of this theory to

polymer chains when they are the object of scissions and recombinations,along the

lines proposed by Faivre and Gardissat [36].

1.3.1 The Rouse model

The Rouse model treats all chains as being independent.Consider a “dead chain” of

N elements at temperature T,where each fragment is subject to a friction force pro-

portional to its instantaneous velocity with friction coeﬃcient ξ and to a random force

modelled as a 3D white noise stochastic process whose statistical properties are given

by the ﬂuctuation-dissipation theorem in terms of T and ξ.The only interactions con-

sidered in the Rouse model are harmonic spring forces with spring constant k between

neighboring beads to ensure the linear connectivity.Under this dynamics,the average

squared distance between adjacent monomers deﬁnes a typical mean distance b given

by b

2

= 3k

B

T/k,where k

B

is the Boltzmann constant.The mean squared end-to-end

distance is simply (N −1)b

2

.

Chain dynamics considered via the end-to-end vector time relaxation function leads

at long times to an exponential decay with the main (Rouse) relaxation time

τ

R

= τ

0

N

2

(1.50)

where τ

0

=

ξb

2

3k

B

Tπ

2

,indicating that the global relaxation time of a chain of size N is

proportional to N

2

.

The way to derive this important result requires the introduction of modes (Rouse

modes),deﬁned as

¯

X

q

(t) =

1

N

N

X

j=1

A

qj

¯

R

j

(t);q = 0,1,2, (N −1) (1.51)

where

¯

R

j

is the coordinate of the j

th

monomer in a chain and

¯

¯

A a square matrix with

elements

A

qj

= cos

qπ

N

(j −

1

2

)

;q = 0,1,2, (N −1);j = 1,2, N (1.52)

18

1.3 Linear viscoelasticity of unentangled micelles

According to this deﬁnition,the mode q = 0 represents the center of mass diﬀusive

motion,while the other (N-1) modes q > 0 are associated to the collective motion of

sub-chains of size N/q.The time autocorrelation of these internal chain modes are

<

¯

X

q

(t)

¯

X

q

(0) >=< X

2

q

> exp

−

t

τ

q

(1.53)

where

τ

q

=

ξb

2

3k

B

Tπ

2

N

q

2

(1.54)

which shows that the Rouse time is the relaxation time associated to the mode q = 1,

while the relaxation time of higher index modes decreases as 1/q

2

.For completeness,

the mode amplitudes are given by

< X

2

q

>=

Nb

2

2π

2

1

q

2

(1.55)

Rouse theory leads to an expression of the time autocorrelation of the chain end-

to-end vector

¯

S =

¯

R

N

−

¯

R

1

<

¯

S(t).

¯

S(0) >= (N −1)b

2

P′

q

τ

q

exp(−t/τ

q

)

P′

q

τ

q

(1.56)

where the prime in the sums means that the latter only involves the odd q mode

indices.This result shows a typical Rouse theory relaxation function:it is a sum

of terms implying the diﬀerent mode relaxation,which explains that at long times,

only the slowest mode survives and the behavior of the time autocorrelation becomes

exponential with characteristic time τ

R

.

1.3.2 The Rouse model implications on the viscous response of a

monodisperse polymer solution

We now discuss the intrinsic shear modulus G(t) of a monodisperse polymer solution

which reﬂects the viscoelastic response of the system in the linear regime (limit of very

small shear rates).The Newtonian shear viscosity is related to the shear modulus

through

η

0

=

Z

∞

0

G(t)dt (1.57)

Rouse theory for a monodisperse solution of “dead” polymers of size N at monomer

number density φ gives [48]

G(t) =

φ

N

k

B

T

N−1

X

q=1

exp(−2t/τ

q

) =

φ

N

k

B

T

N−1

X

q=1

exp(−2tq

2

/τ

R

) (1.58)

19

This important result indicates that,because of the independence of the dynamics

of the various chains in the system,Rouse theory gives for a collective and intensive

quantity like G(t) an expression which is proportional to the solute concentration while

its viscoelastic behavior is strictly identical to the single chain viscoelastic relaxation

process.Therefore,the shear modulus again appears as a sum of terms expressing the

speciﬁc contributions of the various single chain modes in the relaxation process,with

at long times,a simple exponential time decay with characteristic time τ

R

.By time

integration,according to eq.(1.57),the shear viscosity is then given (for N >> 1) by

η

0

=

π

2

12

φk

B

T

N

τ

R

(1.59)

where τ

R

is given in eq.(1.50).

1.3.3 The theory of Faivre and Gardissat and the viscoelasticity of

micelles

The above subsection is dealing with a monodisperse dilute solution of “dead” chains.

The stress relaxation function of a micellar system is in reality the object of a coupling

between the stress relaxation and the scission-recombination process if the time scale of

the second process is shorter than the viscoelastic times scales.In our work,we adapt

the theory proposed by Faivre and Gardissat [36],originally developed to interpret

rheological data of liquid selenium.Faivre and Gardissat [36] proposed a modiﬁcation

of the standard Rouse theory of linear viscosity of a polydisperse polymer system [8]

to incorporate the inﬂuence of the scission events.

If we have a polydisperse system of polymers with normalized weight function W

p

,

the relaxation modulus given by the Rouse model should be given by

G(t) =

∞

X

p=1

W

p

G

p

(t) (1.60)

where G

p

(t) concerns the “polymer” part of chain of size p which,within the Rouse

model,reads (see eq.(1.58))

G

p

(t) = G

0

1

p

p

X

q=1

exp

"

−2

t

τ

0

q

p

2

#

,(1.61)

where p is the polymerization degree,G

0

= φk

B

T a material constant,and τ

0

the

local dynamic relaxation time which depends on the solvent viscosity.Notice that for

20

1.3 Linear viscoelasticity of unentangled micelles

physical reasons we have q ≤ p in the sum of Rouse modes for any upper limit p.As

mentioned earlier,the stress relaxation is a superposition of exponentials,each term

corresponds to the contribution to the relaxation of a particular Rouse mode.All terms

have the same amplitude but decay with a characteristic time

τ

p

q

= τ

0

p

2

q

2

(1.62)

which depends on the p/q ratio which is the number of monomers per wave length for

that particular mode.If bond scission occurs independently and uniformly along the

chain with rate k

s

(per unit time and per bond),the lifetime of a chain of p/q monomers

should be

τ

b

=

q

k

s

p

.(1.63)

The Rouse mode q in the original chain of length p should therefore have a survival

probability which decays in time as exp(−k

s

(p/q)t),hence the idea of multiplying the

contribution of each mode to the relaxation of G(t) by this survival probability to take

into account the eﬀect of bond scissions.This leads to the expression

G

′

p

(t) = G

0

1

p

p

X

q=1

exp

"

−k

s

p

q

t −2

t

τ

0

q

p

2

#

.(1.64)

When the life time τ

b

of the chain of average size is shorter than the internal Rouse

relaxation time of the same chain τ

0

L

2

0

,the viscoelastic response is independent of the

mean chain length L

0

but depends upon a new intrinsic time τ

Λ

which corresponds

to the relaxation of a dynamical unit of size Λ deﬁned by the equality of the Rouse

relaxation time τ

0

Λ

2

and the life time,(k

s

Λ)

−1

:

τ

0

Λ

2

= (k

s

Λ)

−1

.(1.65)

It leads to

Λ = (τ

0

k

s

)

−1/3

(1.66)

and

τ

Λ

= τ

1/3

0

k

−2/3

s

(1.67)

so that equation (1.64) can be rewritten as

G

′

p

(t) = G

0

1

p

p

X

q=1

exp

−

t

τ

Λ

"

p

qΛ

+2

qΛ

p

2

#!

.(1.68)

21

Identifying the number of Rouse beads with the number of beads in present work,using

a known distribution of lengths of the chains of our system (1.18),the weight function

for our equilibrium polymers is

W

p

=

1

L

0

exp

−

p

L

0

.(1.69)

So that G(t) for the equilibrium polymers is now given by

G(t) =

∞

X

p=1

W

p

G

′

p

(t).(1.70)

Note that this expression diﬀers slightly from the Faivre and Gardissat ﬁnal expres-

sion as apparently,in equation (18) of reference [36],they assume a relaxation time for

a segment of p monomers to be half of the usual Rouse time.(Note also that in their

paper,the symbol τ

0

corresponds to half the time τ

0

used in our work).

22

Chapter 2

The mesoscopic model of

worm-like micelles

The main goal our thesis is to exploit simulation techniques to improve our under-

standing of the link between the macroscopic behavior and microscopic aspects of the

structure and the dynamics of self-assembling micellar systems at equilibrium and un-

der shear ﬂow.As it is pragmatically impossible to reach large enough scales of length

and time to determine those macroscopic properties using an atomistic level molecular

dynamics simulation approach,we adopt a mesoscopic model and a Langevin Dynamics

approach as done regularly in the studies on micellar systems.As sketched in ﬁgure 1,

micelles can be represented as linear sequences of Brownian beads which,in addition to

their usual Langevin Dynamics space-time evolution,can either fuse together to form

longer structures or break down into two pieces.The kinetics can be modelled by a

microscopic kinetic Monte-Carlo algorithm which generates new bonds between chain

ends adjacent in space or which breaks existing bonds between adjacent monomers.The

free energy E to creat this new end caps for these micelles becomes an energy penalty

for scission events,i.e.,the creation of two unsaturated chain ends.This scission energy

determines the static propeties and inﬂuences the scission and recombination rates,k

s

and k

r

.In this model,the scission energy E of micelles is controlled by a scission en-

ergy parameter W which is deﬁned as an additive potential term which inﬂuences the

switching probability between bonded and unbounded potentials.The barrier energy

of recombination B also inﬂuences the kinetic rate constants.The advantage of our

model is that the dynamical barrier height B can be taken into account by means of

the attempt frequency ω associated to the Monte Carlo potential swap.If ω is small,

23

successive breakage and recombination events for a given chain can be assumed to be

uncorrelated and the recombination of newly created chain ends will be of standard

mean-ﬁeld type.On the other hand,the (return) probability that two newly created

chain ends recombine immediately must be particulary important at large ω.These

highly correlated “diﬀusion controlled” [12] recombination events do not contribute do

the eﬀective macroscopic reaction rates which determine the dynamics of the system.

The mesoscopic model is based on a standard polymer model with repulsive Lennard-

Jones interactions between all monomer pairs and an attractive FENE (ﬁnitely extensi-

ble nolinear elastic) potential [24] to enforce (linear) connectivity.For the scission and

recombination events,a Monte Carlo procedure is set up to switch back and forth be-

tween the bounded potential and the unbounded potential.This new model is thus at

the same time justiﬁed by its link with a standard polymer model and by the advantage

that the scission/recombination attempt frequency is a control parameter governing dy-

namics,without aﬀecting the thermodynamic and structural properties of the system.

2.1 The potential [19]

We consider a set of micelles consisting of (non-cyclic) linear assemblies of Brownian

particles.Within such a linear assembly,the bond potential U

1

(r) acting between ad-

jacent particles is expressed as the sum of a repulsive Lennard-Jones (shifted and trun-

cated at its minimum) and an attractive part of the FENE type [24].The pair potential

U

2

(r) governing the interactions between any unbounded pair (both intramicellar and

intermicellar) is a pure repulsive potential corresponding to a simple Lennard-Jones

potential shifted and truncated at its minimum.This choice of eﬀective interactions

between monomers implies good solvent conditions.

Using the Heaviside function Θ(x) = 0 or 1 for x < 0 or x ≥ 0 respectively,explicit

expressions (see ﬁgure 2.1) are

U

2

(r) = 4ǫ

(

σ

r

)

12

−(

σ

r

)

6

+

1

4

Θ(2

1/6

σ −r)) (2.1)

U

1

(r) = U

2

(r) −0.5kR

2

ln

1 −[

r

R

]

2

−U

min

−W (2.2)

In the second expression valid for r < R,k = 30ǫ/σ

2

is the spring constant and

R = 1.5σ is the value at which the FENE potential diverges.U

min

is the minimum

value of the sum of the two ﬁrst terms of the second expression (occurring at r

min

=

24

2.2 The Langevin Dynamics

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

r

-10

0

10

20

30

40

50

60

70

80

U

Γ

W

0.96 < r < 1.2

Figure 2.1:Bounded potential U

1

(r) (continuous curve) and unbounded potential U

2

(r)

(dashed line) between a pair of monomers.W is a parameter tuning the energy required

to open the bond.The ﬁgure also shows the Γ region where potential swaps (corre-

sponding to bond scissions or bond recombinations) are allowed during the Brownian

dynamics simulation.The unit of length is σ,and for the energy is ǫ.

0.96094σ for the adopted parameters) while W is a key parameter which corresponds

to the typical energy gain (loss) when an unbounded (bounded) pair is the object of a

recombination (scission).

2.2 The Langevin Dynamics

The Langevin equation describes the Brownian motion of a set of interacting particles.

The action of the ﬂuid is split into a slowly evolving viscous force and a rapidly ﬂuc-

tuating random force.For a free particle in one dimension the equation is expressed

as

m

i

˙

¯v

i

(t) = −ξ¯v(t)

i

+

¯

F

i

(t) +

¯

R

i

(t) (2.3)

where barv

i

is the velocity of particle i with mass m

i

at time t,

¯

F

i

the systematic force,

ξ the friction coeﬃcient,and

¯

R

i

represents the sum of the forces due to the incessant

collision of the ﬂuid molecules.It is regarded as a stochastic force and is satisfying the

25

condition (ﬂuctuation-dissipation theorem) [47]:

< R

iα

(t)R

jβ

(t

′

) >= 2ξk

B

Tδ

ij

δ

αβ

δ(t −t

′

) (2.4)

where the greek-letter subscripts αβ refer to the x,y or z components.

2.2.0.1 Brownian Dynamics (BD)

If we assume that the time scales for momentum relaxation and position relaxation

are well separated,then it is possible to consider only time intervals longer than the

momentumrelaxation times.In the diﬀusive regime the momentumof the particles has

relaxed and equation (2.3) can be modiﬁed by setting to zero the momentum variation

(left hand side of equation (2.4)).Then the “position Langevin equation” is given as

d¯r

i

dt

=

¯

F

i

(t)

ξ

+

¯

R

i

(t)

ξ

(2.5)

Earlier simulations within the present work were executed using an algorithm based on

(2.5) [25].The single BD step of particle i subject to a total force

F

i

is simply

¯r

i

(t +Δt) = ¯r

i

(t) +

¯

F

i

ξ

Δt +

¯

Δ

i

(Δt) (2.6)

where the last term given by

¯

Δ

i

(Δt) =

Z

t+Δt

t

¯

R

i

(t

′

)dt

′

/ξ (2.7)

corresponds to a vectorial random Gaussian quantity with ﬁrst and second moments

given by

< Δ

iα

(Δt) > = 0 (2.8)

< Δ

iαβ

(Δt)Δ

jβ

(Δt) > = 2(

k

B

T

ξ

)Δtδ

αβ

δ

ij

(2.9)

for arbitrary particles i and j and where αβ stand for the x,y,z Cartesian components.

At this stage,it is useful to ﬁx units.In the following,we will adopt the size of

the monomer σ as unit of length,the ǫ parameter as energy unit and we will adopt

ξσ

2

/(3πǫ) as time unit.We also introduce the reduced temperature k

B

T/ǫ = T

∗

,so

that in reduced coordinates (written here with symbol*) the random displacement Δ

∗

iα

is

< Δ

∗

iα

> = 0 (2.10)

< Δ

∗

iα

Δ

∗

jβ

> =

2

3π

Δt

∗

δ

αβ

δ

ij

T

∗

(2.11)

26

2.2 The Langevin Dynamics

which means that each particle,if isolated in the solvent,would diﬀuse with a RMSD of

p

2T

∗

/π per unit of time.In the following,all quantities will be expressed in reduced

coordinates without the ∗ symbol.

2.2.0.2 Langevin Dynamics velocity Verlet scheme

Most of the simulation of the present work are performed using Langevin Dynamics

(LD) velocity Verlet algorithmes based on the work of [50].The explict expressions are

¯r

i

(t +Δt) = ¯r

i

(t) +c

1

¯v

i

(t)Δt +c

2

¯

F

i

(t)

m

Δt

2

+δ¯r

G

i

(2.12)

¯v

i(1/2)

= c

0

¯v

i

(t) +

c

0

c

2

c

1

¯

F

i

(t)

m

Δt +δ¯v

G

i

(2.13)

¯v

i

(t +Δt) = ¯v

i(1/2)

+

1

(γΔt)

1 −

c

0

c

1

¯

F

i

(t +Δt)

m

Δt (2.14)

where the ﬁrst equation updates the position,the second equation updates the “half”

step velocities and the third equation completes the velocity move.γ = ξ/m,the

coeﬃcients for the above equations are:

c

0

= e

−γΔt

(2.15)

c

1

= (γΔt)

−1

(1 −c

0

) (2.16)

c

2

= (γΔt)

−1

(1 −c

1

) (2.17)

c

3

= (γΔt)

−1

1

2

−c

1

(2.18)

where e

γΔt

is approximated by its power expansion for γΔt < 1.Each pair of vec-

torial component of δr

G

,δv

G

,i.e.δr

G

iα

,δv

G

iα

is sampled from a bivariate Gaussian

distribution[49] deﬁned as

ρ

δr

G

iα

,δv

G

iα

=

1

2πσ

r

σ

v

(1 −c

2

rv

)

1/2

×exp

(

−

1

2(1 −c

2

rv

)

δr

G

iα

σ

r

2

+

δv

G

iα

σ

v

2

−2c

rv

δr

G

iα

σ

r

δv

G

iα

σ

v

!)

(2.19)

with zero mean values,variances given by

σ

2

r

=

D

δr

G

iα

2

E

= Δt

2

k

B

T

m

(γΔt)

−1

2 −(γΔt)

−1

3 −4e

−γΔt

+e

−2γΔt

(2.20)

σ

2

v

=

D

δv

G

iα

2

E

=

k

B

T

m

1 −e

−2γΔt

(2.21)

27

and the correlation coeﬃcient c

rv

determined by

c

rv

=

δr

G

iα

δv

G

iα

= Δt

k

B

T

m

(γΔt)

−1

1 −e

−ξΔt

2

1

σ

r

σ

v

(2.22)

Each pair of cartesian components δr

G

iα

and δv

G

iα

are obtained by the appropriate Gaus-

sian distribution according to:

δr

iα

= σ

r

η

1

(2.23)

δv

iα

= σ

v

c

rv

η

1

+η

2

p

1 −c

2

rv

(2.24)

where η

1

and η

2

are two independent random numbers with Gaussian distribution of

zero average and unit variance.

2.2.1 Nonequilibrium LD technique

In this sub-section we brieﬂy present the technique adopted for imposing the shear ﬂow

onto the system.We limit the study to a stationary,isovolumic and homogeneous planar

Couette ﬂow characterized by a velocity ﬁeld ¯u(r) =

¯

¯

k

T

r as illustrated in ﬁgure 2.2,

where the velocity gradient

¯

¯

k is constant in space and time.

Figure 2.2:Sketch of planar Couette ﬂow.

For a planar Couette ﬂow in the x direction with a shear along the y axis,the

velocity gradient k is deﬁned as

¯

¯

k =

0 0 0

˙γ 0 0

0 0 0

where ˙γ is shear rate.With the imposition of the solvent velocity ﬁeld,the equation of

motion becomes

m

i

˙

¯v

i

(t) = −mξ

i

(¯v

i

(t) − ¯u(r

i

)) +

¯

F

i

(t) +

¯

R

i

(t) (2.25)

28

2.3 Monte-Carlo procedure

A modiﬁcation of the periodic boundary conditions proposed by [51] has been adopted

Figure 2.3:Sketch of shear boundary conditions for planar Couette ﬂow.

for establishing the shear ﬂow.The scheme used is shown in ﬁgure 2.3.In essence,the

inﬁnite periodic system is subjected to a uniform shear in xy plane.The box above

is moving at a speed ˙γL in the positive x direction.The box below moves at a speed

˙γL in the negative x direction.When a particle in the primary box with coordinates

(x,y) crosses a horizontal border one of its images enters from the opposite side at

x

′

= x − ˙γtL

y

with velocity v

′

= v − ˙γL

y

2.3 Monte-Carlo procedure

The bonding network is itself the object of random instantaneous changes provided by

a Monte-Carlo (MC) algorithm [19] which is built according to the standard Metropolis

scheme [49].

The probability P

m,n

to go from a bounding network m to a diﬀerent one n is

written as

P

m,n

= A

trial

m,n

P

acc

m,n

(2.26)

where A

trial

m,n

is the trial probability to reach a new bounding network n starting from

the old one m,within a single MC step.This trial probability is chosen here to be

symmetric as usually adopted in Metropolis Monte Carlo schemes,and is chosen to be

29

diﬀerent from zero only if both bounding networks n and m diﬀer by the status of a

single bond,say the pair of monomers (ij).To satisfy microreversibilty,the acceptance

probability P

acc

m,n

for the trial (m−→n) must be given by

P

acc

m,n

= Min[1,exp (−

(U({r},n) −U({r},m))

k

B

T

)].(2.27)

In the present case,as a single pair (ij) changes its status,the acceptance probability

takes the explicit form

P

acc

m,n

= Min[1,exp(−

(U

2

(r

ij

) −U

1

(r

ij

))

k

B

T

)] (2.28)

P

acc

m,n

= Min[1,exp(−

(U

1

(r

ij

) −U

2

(r

ij

))

k

B

T

)] (2.29)

for bond scission and bond recombination respectively.

The way to specify the trial matrix A

trial

m,n

starts by deﬁning a range of distances,

called Γ and deﬁned by 0.96 < r < 1.20 within the range r < R.For r ∈ Γ,a

change of bounding is allowed as long as the two restricting rules stated above are

respected.Consider the particular conﬁguration illustrated by ﬁgure 2.4 where M=7

monomers located at the shown positions,are characterized by a connecting scheme

made explicit by representing a bounding potential by a continuous line.The dashed

line between monomers 5 and 6 represents the changing pair with distance r ∈ Γ which

is a bond U

1

(r) in conﬁguration m but is just an ordinary intermolecular pair U

2

(r) in

conﬁguration n.

For further purposes,we have also indicated in ﬁgure 2.4 by a dotted line all pairs

with r ∈ Γ which are potentially able to undergo a change from a non bounded state

to a bounded one in the case where the (56) pair,on which we focus,is non bounded

(state n).Note that the bond (35) is not represented by a dotted line eventhough the

distance is within the Γ range:a bond formation in that case is not allowed as it would

lead to a cyclic conformation.Note also that in state n,monomer 5 could thus form

bonds either with monomer 2 or with monomer 6 while monomer 6 can only form a

bond with monomer 5.

We now state the algorithm and come back later on the special (m−→n) transition

illustrated in ﬁgure 2.4.

During the LDdynamics,with an attempt frequency ω per armand per unit of time,

a change of the chosen arm status (bounded U

1

(r) to unbounded U

2

(r) or unbounded

30

2.3 Monte-Carlo procedure

to bounded) is tried.If it is accepted as a “trial move” of the bounding network,it

obviously implies the modiﬁcation of the status of a paired arm belonging to another

monomer situated at a distance r ∈ Γ from the monomer chosen in the ﬁrst place.

The trial move goes as follows:a particular arm is chosen,say arm 1 of monomer

i,and one ﬁrst checks whether this arm is engaged in a bounding pair or not.

• If the selected arm is bounded to another arm (say arm 2 of monomer j) and the

distance between the two monomers lies within the interval r

ij

∈ Γ,an opening

is attempted with a probability 1/(N

i

+1),where the integer N

i

represents the

number of monomers available for bonding with monomer i,besides the monomer

j (N

i

is thus the number of monomers with at least one free arm whose distance

to the monomer carrying the originally selected arm i lies within the interval Γ,

excluding from counting the monomer j and any particular arm leading to a ring

closure).If the trial change consisting in opening the (ij) pair is refused (either

because the distance is not within the Γ range or because the opening attempt

has failed in the case N

i

> 0),the MC step is stopped without bonding network

change (This implies that the LD restarts with the (ij) pair being bonded as

before).

• If the selected arm (again arm 1 of monomer i) is free from bonding,a search is

made to detect all monomers with at least one armfree which lie in the “reactive”

distance range r ∈ Γ from the selected monomer i (Note that if monomer i is a

terminal monomer of a chain,one needs to eliminate from the list if needed,the

other terminal monomer of the same chain in order to avoid cyclic micelles conﬁg-

urations).Among the monomers of this “reactive” neighbour list,one monomer

is then selected at random with equal probability to provide an explicit trial

bonding attempt between monomer i and the particular monomer chosen from

the list.Note that if the list is empty,it means that the trial attempt to create a

new bond involving arm 1 of monomer i has failed and no change in the bonding

network will take place.

In both cases,if a trial change is proposed,it will be accepted with the probability

P

acc

m,n

deﬁned earlier.If the change is accepted,LDwill be pursued with the newbonding

scheme (state n) while if the trial move is ﬁnally rejected,LD restarts with the original

bonding scheme corresponding to state m.

31

Figure 2.4:Exemplary conﬁguration of a 7 monomers systemin state n where monomers

3,4 and 5 form a trimer and monomers 6 and 7 a dimer.All pairs of monomers with

mutual distances within the Γ region are indicated by a dotted or a dashed line.In the

text,we consider the Monte Carlo scheme for transitions between states n and m which

only diﬀer by the fact that in state n and mthe 5-6 pair is respectively open or bounded.

The n →m transition corresponds to the creation of a pentamer by connecting a dimer

and a trimer while the m → n transition leads to the opposite scission.The cross

symbol on link 3-5 indicates that in state n,when looking to all monomers which could

form a new link with monomer 5,monomer 3 is excluded because it would lead to a

cyclic polymer which is not allowed within the present model.

Coming back to ﬁgure 2.4,we now show that the MC algorithm mentioned above

garantees that the matrix A

trial

mn

is symmetric,an important issue as it leads to the

micro-reversibility property when combined with the acceptance probabilities described

earlier.Let us deﬁne as P

arm

= 1/2N the probability to select a particular arm,a

uniform quantity.

If conﬁguration m with pair (56) being “bounded” is taken as the starting con-

ﬁguration,the number of available arms to form alternative bonds with monomer 5

and monomer 6 are respectively N

5

= 1 and N

6

= 0.Therefore,applying the MC

rules described above,the probability to get conﬁguration n where the pair (56) has to

be unbounded is given by the sum of probabilities to arrive at this situation through

selection of the arm of monomer 5 engaged in the bond with monomer 6 or through

selection of the arm of monomer 6 engaged in a bond with monomer 5.This gives

A

trial

m,n

= P

arm

∗

1

N

5

+1

+P

arm

1

N

6

+1

=

3

2

∗ P

arm

(2.30)

If conﬁguration n with pair (56) being “unbounded” is taken as the starting conﬁg-

uration,the application of the MC rules lead to the probability to get conﬁguration m

32

2.3 Monte-Carlo procedure

where the pair (56) has to be bounded is given by the sum of probabilities to arrive at

this situation through selection of the free arm of monomer 5 (which has two bounding

possibilities,namely with monomers 2 and 6) or through selection of the free arm of

monomer 6 which can only form a bond with monomer 5.This gives

A

trial

n,m

= P

arm

∗

1

2

+P

arm

=

3

2

∗ P

arm

(2.31)

showing the required matrix symmetry.

33

34

Chapter 3

Equilibrium Properties

We have exploited the model introduced in chapter II in a series of Langevin Dynam-

ics (LD) simulations at equilibrium with diﬀerent bonding energy parameter W and

diﬀerent number density φ,and hence,in this chapter,we will present the resulting

equilibrium static and dynamic properties of cylindrical micelles.

Within the static properties,the theoretical prediction of the chain size distribution

has been given by Cates [2] and tested in great detail by Monte Carlo simulation by

Wittmer et al.[13,52] who also investigated conformational properties of chains at

equilibrium.For static properties,our aim is thus mainly to check that our results on

a diﬀerent model are compatible with the analysis of the previous works [13,52].

The two main parameters governing the static properties are the monomer density

φ and the end cap energy E.Our choice of these two parameters is set up with the

aim to simulate three thermodynamic states corresponding to a dilute solution and two

semi-dilute solutions at the same φ but diﬀerent E,leading to the system with two

average chain lengths ∼ 56 and ∼ 150.

For static properties,the distribution of chain lengths,the gyration radius,and

the end-to-end distance versus chain length will be analyzed and compared with pre-

vious studies.We have also investigated g

ee

(r) the pair correlation function of end

monomers and P(r) the distribution of bond lengths which are quantitative pertinent

in the microscopic formulation of the kinetic rate constants.

The dynamic properties of the micelles which are given and interpreted in this

chapter form the core of the work.The detailed trajectory of diﬀusing micelles which

are continuously breaking and recombing allows us to analyze the microscopic origin of

rate constants in terms of structural features (e.g.chain end pair correlation function)

35

and dynamic quantities to be related to the statistics of life times of a newly created

chain end.For the latter,we show that the Poisson statistics dominates at long times

while a fraction of correlated recombinations happen at short times.Exploiting these

microscopic features,we characterize the macroscopic scission energy E and the barrier

of recombination B and estimate their values for various state points investigated.

Some macroscopic dynamic properties are then studied with an accent on the mod-

iﬁcation of various dynamic relaxation processes due to the scission-recombination pro-

cess.We investigate the monomer diﬀusion and the stress relaxation function.Finally,

we perform a T-jump experiment in order to point out that the previously estimated

macroscopic kinetic constants are indeed the key parameters governing the relaxation

of the chain length distribution.

3.1 Static properties

The main aim of this section is to test the model of chapter II by comparing the struc-

tural properties,including the average chain length,the distribution of chain lengths

and the conformations of the chains,with the theory [9,27] and previous simulation

works [13,52].

3.1.1 List of simulation experiments and chain length distribution

The model is studied at three state points.The number of monomers,the number

density φ,the energy parameter W,and the attempt frequency ω per arm are chosen

for

1.A solution at the number density φ = 0.05 and an energy parameter W = 8.

The number of monomers is M=1000.The attempt frequencies of bond scis-

sion/recombination per arm ω are 0.1,0.5,1 and 5.This choice will be shown to

lead to a dilute solution.

2.A solution at the number density φ = 0.15 and an energy parameter W = 10.

The number of monomers is M=1000.The attempt frequencies of bond scis-

sion/recombination per arm ω are 0.1,0.5,1 and 5.(will be shown to be a

semi-dilute solution)

36

3.1 Static properties

3.A solution at the number density φ = 0.15 and an energy parameter W = 12.

The number of monomers is M=5000.The attempt frequencies of bond scis-

sion/recombination per arm ω are 0.02,0.06,0.1,0.5 and 1 (will be shown to be

a semi-dilute solution).

Each system evolves according to the Langevin Dynamics algorithm with time step

Δt = 0.005 and is subject to randomtrials of bond scission/recombination with the arm

attempt frequency ω.All the experiments and the results of static properties which

include the average chain length L

0

,the mean square end-to-end vector,the radius

of gyration and L

0

/L

∗

,the number of blobs in a chain of length L

0

,are listed in the

table 3.1.Where

R

2

and

R

2

g

are deﬁned as

R

2

L

0

=

L

0

−1

X

n=1

L

0

−1

X

m=1

h¯r

n

¯r

m

i (3.1)

R

2

g

L

0

=

1

2L

2

0

L

0

X

n=1

L

0

X

m=1

(

¯

R

n

−

¯

R

m

)

2

,(3.2)

where ¯r

n

=

¯

R

n+1

−

¯

R

n

.And L

∗

is deﬁned by equation (1.21) (also see section 3.1.2).

As shown in the table,we observed that all the static properties are independent

of ω and therefore,all data can be averaged over all ω values.

3.1.1.1 The dilute case

From Table 3.1,it can be observed that the ﬁrst state point experiment (W = 8,φ =

0.05) is a dilute solution since its average chain length L

0

= 11.48(1) is much smaller

than its crossover value at the monomer number density as calculated by equation (1.20),

L

∗

= 50.5.Dilute solution conditions are conﬁrmed by the chain length distribu-

tion shown in ﬁgure 3.1.For dilute conditions,a distribution given by (1.30) is ex-

pected [13,52].We ﬁt our data with a single parameter B

1

of function (1.30),where

L

0

is given its computed average value and where γ is given its expected value,γ = 1.165

[13].The ﬁt gives B

1

= 1.08.This curve is signiﬁcantly better than the simple expo-

nential distribution expected for ideal or semi-dilute chain.If γ is left as a second free

parameter in the ﬁt,it gives γ = 1.161 which is also very close to its expected value.

37

Table 3.1:List of simulation experiments and the values of static properties.W is the

scission energy parameter,φ the number density,T

s

the total simulation time,ω the

attempt frequency,L

0

the average chain length,

q

hR

2

i

L

0

the end-to-end distance of

the average chain,and

q

R

2

g

L

0

its radius gyration.L

0

/L

∗

is the ratio of the average

chain length over the blob length L

∗

W

φ

T

s

ω

L

0

q

hR

2

i

L

0

q

R

2

g

L

0

L

0

/L

∗

8

0.05

2.5 ∗ 10

5

0.1

11.48(6)

4.93(2)

1.9(2)

0.2

8

0.05

2.5 ∗ 10

5

0.5

11.46(2)

4.92(1)

1.9(1)

0.2

8

0.05

2.5 ∗ 10

5

1.

11.50(3)

4.94(1)

1.9(1)

0.2

8

0.05

2.5 ∗ 10

5

5.

11.51(2)

4.95(1)

1.9(1)

0.2

10

0.15

2.5 ∗ 10

5

0.1

56.2(6)

12.2(1)

4.8(2)

4.7

10

0.15

2.5 ∗ 10

5

0.5

56.2(1)

12.4(1)

4.9(3)

4.7

10

0.15

2.5 ∗ 10

5

1.

56.2(2)

12.3(1)

4.9(3)

4.7

10

0.15

2.5 ∗ 10

5

5.

56.6(2)

12.4(1)

4.9(4)

4.7

12

0.15

6.25 ∗ 10

5

0.02

151(4)

20.9(5)

8.3(4)

12.6

12

0.15

4.5 ∗ 10

5

0.06

153(4)

21.5(7)

8.5(5)

12.6

12

0.15

4.5 ∗ 10

5

0.1

150.7(5)

21.2(2)

8.4(3)

12.6

12

0.15

3 ∗ 10

5

0.5

150.3(5)

20.9(1)

8.4(2)

12.5

12

0.15

3 ∗ 10

5

1

150.4(6)

20.92(3)

8.5(4)

12.5

3.1.1.2 The semi-dilute case

Both the second state point (W = 10,φ = 0.15) and the third state point (W =

12,φ = 0.15) experiments are found to be in semi-dilute regime,since their average

chain lengths are 56.4(1) and 151.4(4),respectively,which are several times larger

than their crossover value L

∗

≈ 12 at φ = 0.15 according to equation (1.20).Semi-

dilute solution conditions are conﬁrmed by the observation of a simple exponential

distribution of chain lengths.In ﬁgure 3.2 and ﬁgure 3.3,we show,for the second

and the third state point experiments respectively,the agreement of our data with the

expected simple exponential distribution (1.34).

3.1.2 Chain length conformational analysis

In this subsection we are interested in the conformational properties of micelles in

equilibrium and studied as a function of chain size within our polydisperse system.

We have calculated the mean square end-to-end distance and the mean square radius

of gyration < R

2

> (L) and < R

2

g

> (L) averaged over subsets of chains of length L.

38

3.1 Static properties

0 1 2 3 4 5 6 7 8 9 10

L/L

0

10

-8

10

-7

10

-6

10

-5

10

-4

10

-3

c0(L)

Figure 3.1:Distribution of chain lengths for the dilute case in good solvent.The data

(squares) are ﬁtted using equation (1.30) with the single parameter B

1

and imposed

value γ = 1.165 (continuous curve).The ﬁt gives B = 1.08.The dashed line shows

the simple exponential distribution c

0

(L) ∝ exp(−L/L

0

) which does not ﬁt the data.

Figure 3.4 and ﬁgure 3.5 are the results for the dilute state point and the two semi-dilute

state points experiments respectively.As shown in ﬁgure 3.4 relative to the dilute case,

the squares and circles represent our data of < R

2

> and < R

2

g

> respectively.With

the solid line and the dashed line,we indicate,for long chains (L 25),standard

power law scaling L

2ν

with ν = 0.588.The data show a reasonable agreement of this

asymptotic regime in the range of application.

The chains in the semi-dilute system are expected to behave as ideal chains for

L ≫ L

∗

≈ 12 where L

∗

is estimated from equation.(1.20) with φ = 0.15.Figure 3.5

shows the

R

2

and

R

2

g

of the two semi-dilute cases.Results of

R

2

and

R

2

g

for the two cases,are superimposed.We indicate for the long chains the ideal chain

conformation

R

2

and

R

2

g

∝ L.The agreement of the ﬁtting lines with simulation

data appears to start at L 60.

The conformational properties of the chain in the two semi-dilute cases are found

39

0 1 2 3 4 5 6 7

L/L

0

10

-8

10

-7

10

-6

10

-5

10

-4

c0(L)

Figure 3.2:Distribution of chain lengths for the state point W = 10,φ = 0.15 (ac-

cumulated over all ω values).The data (circles) are ﬁtted very well by the simple

exponential function,equation (1.34),with imposed average chain length L

0

= 56.4.

The Dashed line shows the function exp(−1.165L/L

0

) which does not ﬁt the data.

to be identical as expected.The size R of a chain of length L > L

∗

is

R = bL

∗

ν

L

L

∗

0.5

(3.3)

where b is the monomer size and ν is the good solvent scaling exponent while L

∗

(function of φ only) is the same in both cases.

3.1.3 Pair correlation function of chain ends and the distribution of

bond length

Equilibrium polymers are polydisperse polymers endowed with scission and recombi-

nation processes.Whereas a scission can happen between any bounded pair,a recom-

bination may happen only with two chain ends for linear chains.Thus it is interest-

ing to study the spatial distribution of the chain ends and the bond length distribu-

tion.For the dilute case the population of bonds ready to open within the Γ range

(−0.96 < r < 1.2) is < N

1Γ

>= 552 and represents 61% of the bonded pairs < N

1

>,

while the population of free arm pairs ready to close,again within the same Γ range,is

40

3.1 Static properties

0 1 2 3 4 5 6

L/L

0

10

-8

10

-7

10

-6

10

-5

c0(L)

Figure 3.3:Distribution of chain lengths for the state point W = 12,φ = 0.15 (ac-

cumulated over all ω values).The data (circles) are ﬁtted very well by the simple

exponential function,equation 1.34,with imposed average chain length L

0

= 151.4.

The Dashed line shows the function exp(−1.165L/L

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